Mathematical induction is one of the most important proof techniques in math.
This technique is like a chain of objects in the picture attached.

Actually, we use this technique in order to prove that a statement p(n) is true for any natural number n.

This technique is divided into two parts:

First part, we have to show that p(0) is true, that is the statement is true for n=0 (actually not especially for n=0, but for the lower bound of n that is mentioned in the theorem).

In the second part, we assume that p(n) is true and then we prove that p(n+1) is also true.

From these two steps, we can easily see that p(0) is true, which implies that p(1) is true, which yields also that p(2) is true, and so on we get p(n) is true for any natural number n.

This way of thinking is actually brilliant, and it is one of the issues that mix logic with math.